Multilevel modeling in R
Tom Dunn and Thom Baguley, Psychology, Nottingham Trent
University
Thomas.Baguley@ntu.ac.uk
1. Models for repeated measures or
clustered data
2
Repeated measures ANOVA
Usual practice is psychology is to analyze repeated
measures data using ANOVA:
One-way independent measures
Yij = m + t j + e ij
åt
j
=0
One-way repeated measures
Yij = m + p i + t j + e ij
åt
j
=0
åp
i
=0
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Limitations of standard approaches
e.g., repeated measures ANOVA
- sphericity or multi-sample sphericity assumptions
- dealing with non-orthogonal predictors
e.g., time-varying covariates in RM ANOVA
- dealing with missing values
- treating items as fixed effects (e.g., Clark, 1973)
- the problem of categorization
- the problem of aggregation/disaggregation
4
Avoid repeated measures regression!
Repeated measures regression is one attempt to deal
with limitations of ANOVA such as non-orthogonal
predictors
e.g., using manual dummy coding
(Lorch & Myers, 1990; Pedhazur, 1982)
- data hungry (each indicator requires 1 df)
- assumes sphericity; fixed effects
- less flexible, powerful than multilevel models
(Misangyi et al., 2006)
5
Multilevel models with random intercepts
A random intercept model has predictors with fixed
effects only:
e.g., fixed
random
Yij = b0ij + b1 X1ij + b2 X2ij
b0ij = b0 + u j + eij
… or combined in single equation:
Yij = b0 + b1 X1ij + b2 X2ij + u j + eij
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7
Random effects in multilevel models
In the random intercept model the individual differences at level
2 are (like the random error at level 1) assumed to have a normal
distribution:
Individual differences in the effect of a predictor can also be
modeled this way:
Yij = b0ij + b1 X1ij + b2 j X2ij
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Example - voice pitch (1)
How is male voice pitch is related to subjective attractiveness of
a female face?†
- 30 male participants
- 32 female faces
- ratings of attractiveness (1-9) in 2 contexts
- potential time-varying covariates (e.g., baseline measure)
Classical ANOVA:
i) treats participants (but not faces) as random sample
ii) can't incorporate time-varying covariates
iii) aggregates data (effective n = 30 per context)
† Data from Dunn, Wells and Baguley (in prep) and in Baguley (2012)
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Example - voice pitch (1) contd.
library(nlme)
pitch.ri <- lme(pitch ~ base + attract, random=~ 1|Participant, data=pitch)
summary(pitch.ri)
Linear mixed-effects model fit by REML
Data: pitch
Log-restricted-likelihood: -7061.513
Fixed: pitch ~ base + attract
(Intercept)
89.5863997
base
0.2091518
attract
0.4654556
Random effects: Formula: ~1 | Participant
(Intercept) Residual
StdDev:
13.05003 9.214851
Number of Observations: 1920 Number of Groups: 30
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Modeling covariance matrices 1
In a two level random-intercept model the covariance structure is:
éës u2 ùû
éës e2 ùû
This, in effect, assumes a form of compound symmetry of the
repeated measures (with equal variances and covariances all zero)
12
Modeling covariance matrices 2
In a two level random-slope model with one predictor random at level 2 the
covariance matrix is:
é s2
ê u0
ê su
01
ë
s u2
1
ù
ú
ú
û
éës e2 ùû
In a repeated measures design this models the individual differences in the
effect of the predictor (as well as its covariance with the intercept).
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Unstructured covariance matrices
In repeated measures it is possibly to have an unconstrained covariance
matrix at the participant level (usually level 2). This is an example for four
measurement occasions:
é
ê
ê
ê
ê
ê
ê
ë
s u2
1
su
s u2
su
su
su
su
12
13
14
2
23
s u2
24
su
3
24
s u2
4
ù
ú
ú
ú
ú
ú
ú
û
In this kind of unstructured matrix there are no assumptions about the
form of the matrix (e.g., sphericity, compound symmetry or multisample
sphericity)
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Example - random effect of voice pitch (2)
Attractiveness might not have a fixed effect (in fact it is more
likely that it varies between people)
lme(pitch ~ base + attract, random=~ attract|Participant, data=pitch)
AIC
BIC
logLik
14133.03 14160.82 -7061.513
Random effects: Formula: ~1 | Participant
(Intercept) Residual
StdDev:
13.05003 9.214851
Fixed effects: pitch ~ base + attract
Value Std.Error
DF
t-value p-value
(Intercept) 89.58640
5.537340 1888 16.178598
0
base
0.20915
0.044474 1888
4.702804
0
attract
0.46546
0.104619 1888
4.449042
0
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2. Estimation and inference
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Estimation in multilevel models
Estimation is iterative and usually uses maximum likelihood (as
with logistic regression):
- IGLS or FML (iterative generalized least squares)
- RIGLS or RML (restricted maximum likelihood estimation)
- Parametric bootstrapping
- Non-parametric bootstrapping
- MCMC (Markov chain Monte Carlo methods)
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Comparing models
Confidence intervals and tests
- deviance (likelihood ratio) tests
(-2LL or change in -2 log likelihood has approximate χ2 distribution)
- Wald tests and CIs
(estimate/SE has approximate z distribution)
Information criteria
- AIC, BIC or (MCMC derived) DIC
(-2LL with a penalty for number of parameters)
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Accurate inference …
- for standard repeated measures ANOVA models it is
possible to use t and F statistics
- if a complex covariance structure (anything other
than compound symmetry) or unbalanced model is
used then inference is problematic owing to:
a) difficulty estimating the error df
b) boundary effects (for variances)
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Possible solutions
- asymptotic approximations (in large samples)*
- corrections such as the Kenwood-Rogers
approximation (e.g., using pbkrtest)
- bootstrapping*
- MCMC estimation (e.g., using lme4 or MCMCglmm)*
* e.g., see Baguley (2012) for examples
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Requirements for accurate estimation
Centering predictors
- essential to use appropriate centering strategy in random slope models
see Enders & Tofighi (2007)
Nested versus fully-crossed structures
- many experimental designs in psychology are fully crossed
see Baayen et al. (2008)
Estimation, sample size and bias
- sample size at highest level of model is crucial
see Hox (2002), Maas & Hox (2005)
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Nested versus fully crossed structures
In nested structures lower level units occur in only one
higher level unit
e.g., children in schools
In fully crossed structures ‘lower’ level units are
observed within all ‘higher’ level units
e.g., same 32 faces used for all 30 participants
Baayen et al. (2006) argue that many researchers
incorrectly model fully crossed structures as nested
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Example - fully crossed model (3)
detach(package:nlme) ; library(lme4)
lmer(pitch ~ base + attract + (1|Participant) + (1|Face), data=pitch) †
Formula: pitch ~ base + attract + (1 | Participant) + (1 | Face)
AIC
BIC logLik deviance REMLdev
14134 14167
-7061
14118
14122
Random effects:
Groups
Face
Name
(Intercept)
Variance
Std.Dev.
0.44417
0.66646
Participant (Intercept) 171.72946 13.10456
Residual
84.47292
9.19092
Number of obs: 1920, groups: Face, 32; Participant, 30
Fixed effects:
Estimate Std. Error t value
(Intercept) 90.03032
5.54083
16.249
base
0.20543
0.04447
4.620
attract
0.45910
0.11229
4.088
lmer(pitch ~ base + attract + (attract|Participant) + (1|Face), data=pitch)
† Assumes that the fully crossed structure is correctly coded in the data set
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Advantages of multilevel approaches
- often greater statistical power (e.g., for RM ANOVA)
- multiple random factors (nested or crossed)
- copes with non-orthogonal predictors
- copes with time-varying covariates
- assumes missing outcomes are MAR (not MCAR)
- explicitly models variances and covariances
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